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arXiv:1001.3747 · Physical Review E

Quantum phase transition in a disordered long-range transverse Ising antiferromagnet

Anjan Kumar Chandra, Jun‐ichi Inoue, Bikas K. Chakrabarti

Published Feb 1, 2010• 9 citations
arXivHTMLPDF

Abstract

We consider a long-range Ising antiferromagnet put in a transverse field (LRTIAF) with disorder. We have obtained the phase diagrams for both the classical and quantum cases. For the pure case applying quantum Monte Carlo method, we study the variation in order parameter (spin correlation in the Trotter direction), susceptibility, and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We discuss generally the phase diagram for the same LRTIAF model with perturbative Sherrington-Kirkpatrick-type disorder. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the pure LRTIAF system, an infinitesimal SK spin-glass disorder is enough to induce a stable glass order in the LRTIAF. This glass order eventually gets destroyed as the thermal or quantum fluctuations are increased beyond their threshold values and the transition to paramagnetic phase occurs. Analytical studies for the phase transitions are discussed in details in each case. These transitions have been confirmed by applying classical and quantum Monte Carlo methods. We show here that the disordered LRTIAF has a surrogate incubation property of the SK spin glass phase.

DOI
10.1103/physreve.81.021101

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We consider a long-range Ising antiferromagnet put in a transverse field (LRTIAF) with disorder. We have obtained the phase diagrams for both the classical and quantum cases. For the pure case applying quantum Monte Carlo method, we study the variation in order parameter (spin correlation in the Trotter direction), susceptibility, and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We discuss generally the phase diagram for the same LRTIAF model with perturbative Sherrington-Kirkpatrick-type disorder. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the pure LRTIAF system, an infinitesimal SK spin-glass disorder is enough to induce a stable glass order in the LRTIAF. This glass order eventually gets destroyed as the thermal or quantum fluctuations are increased beyond their threshold values and the transition to paramagnetic phase occurs. Analytical studies for the phase transitions are discussed in details in each case. These transitions have been confirmed by applying classical and quantum Monte Carlo methods. We show here that the disordered LRTIAF has a surrogate incubation property of the SK spin glass phase.
We consider a long-range Ising antiferromagnet (LRIAF) put in a transverse field. Applying quantum Monte Carlo method, we study the variation of order parameter (spin correlation in Trotter time direction), susceptibility and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We also discuss the phase diagram for the Sherrington-Kirkpatrick (SK) model with the same LRIAF bias, also in presence of a transverse field. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the system, an infinitesimal SK spin glass disorder is enough to induce a stable glass order in the antiferromagnet. This glass order eventually gets destroyed as the thermal or quantum fluctuations increased beyond their threshold values and the transition to para phase occurs. Indications of this novel phase transition are discussed. Because of the presence of full frustration, this surrogate property of the LRIAF for incubation of stable spin glass phase in it (induced by addition of a small disorder) should enable eventually the study of classical and quantum spin glass phases by using some perturbation theory with respect to the disorder.
In this work we study the effects of quantum fluctuations in a disordered magnetic system. We consider a cluster-spin Ising model with antiferromagnetic short range interactions (J0) and disordered long range interactions (J) between clusters, where the system is under the influence of a transverse magnetic field Γ. The analytic treatment of the model is done by the replica method and Trotter formalism, in which a cluster mean field, replica symmetric and static approximations are used. This allows us to obtain an effective model of a single cluster that is exactly diagonalized. Thermodynamic quantities like order parameter and free energy are obtained then, the phase diagrams are analyzed. They are shown in temperature versus J0=J phase diagrams for some intensities of Γ. With Γ = 0, when we decrease the temperature (for lower intensities of J0=J) a continuous phase transition between the paramagnetic and cluster spin glass state (CSG) is found. When we increase the ratio J0=J, the system presents the antiferromagnetic phase (AF). The CSG state and the AF phase are separated by first order transitions. In the intermediate region of disorder, there exist a small interval in the AF-PM transition that is first order. When Γ is increased, the transitions PM-AF and PM-VSC decrease toward critical points in the lower and higher disorder, respectively. It is worth to note, in the intermediate range of disorder, that the transition CSG-AF disappears, letting a PM phase between them, even for zero temperature. Besides that, the first-order region, in the PM-AF transition, turns into a second order phase transition, changing its criticality in the low temperature regime. Our results show that quantum fluctuations affects the phase transitions of the model being capable to suppress and/or change his criticality. The present work suggests a model and a mechanism of the change in the criticality away from the quantum critical point, helping the investigation of this interesting and recent topic.
We introduce the transverse Ising model as a prototype for discussing quantum phase transitions. Mean field theory and its application to superconductivity by BCS are discussed, as well as real space renormalization group techniques in one dimension. Next, we introduce the Suzuki-Trotter formalism to show the correspondence between d-dimensional quantum systems and (d + 1)-dimensional classical systems. We then discuss transverse Ising spin glass models, namely the Sherrington-Kirkpatrick (SK) model, the Edwards-Anderson (EA) model and the ±J model. We briefly discuss the mean field, exact diagonalization and quantum Monte Carlo results for their phase diagrams. We discuss the question of replica symmetry restoration in quantum spin glasses due to the possibility of tunneling through the barriers. Finally, we discuss the quantum annealing technique and indicate its relationship with replica symmetry restoration in quantum spin glasses.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
We study the critical behavior of the Sherrington-Kirkpatrick model in transverse field (at finite temperature) using Monte Carlo simulation and exact diagonalization (at zero temperature). We determine the phase diagram of the model by estimating the Binder cumulant. We also determine the correlation length exponent from the collapse of the scaled data. Our numerical studies here indicate that critical Binder cumulant (indicating the universality class of the transition behavior) and the correlation length exponent cross over from their "classical" to "quantum" values at a finite temperature (unlike the cases of pure systems, where such crossovers occur at zero temperature). We propose a qualitative argument supporting such an observation, employing a simple tunneling picture.
We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at T=0T=0T=0.